In science fiction, it is pretty standard fare to introduce some form of faster-than-light communication or travel. After all, space is big, and you can't write your swashbuckling Hornblower-in-space novel if you have to wait for a generation ship to crawl painfully slowly between the nearest stars, much less try to cross a galaxy.
However, faster-than-light communication (which includes travel) breaks something very fundamental about physics, something that is often ignored by sci-fi, and difficult for non-physicists to understand. If you allow faster-than-light (FTL), then you break causality: you are allowing time-travel. One pithy way of saying this is:
The Universe has picked relativity and causality, it seems. Thus, we cannot travel or communicate faster than light.
But why is this? Why does FTL imply time travel. To demonstrate this, it's handy to draw some diagrams. We're going to work with "spacetime diagrams." They look like this:
Here I'm trying to draw all four dimensions of the Universe: three space and one time. Now, I can't draw four dimensions. I can't even really draw three (it's a 2D screen, after all). So I've suppressed two space dimensions, drawing all of space as just a line. It won't matter much for what I'm trying to do, but it's good to keep that in mind.
With that in mind, I'm showing here my spacetime diagram: I'm stationary at the center, and so I see time tick forward "orthogonal" (perpendicular) to the space directions around me. As you'll see, other people have different spacetime diagrams, and different time and space axes, relative to me. That's the point of relativity, after all.
The special thing about relativity is that everyone measures the speed of light to be the same. We show this in a spacetime diagram by saying that every spacetime diagram has light traveling at 45 degrees relative to the time axis. Light travels on lines that are called "null."
Here I'm showing the null lines of light emitted from an event at the time I'm calling t=0 (when the time and space axis cross). Remember I'm suppressing most of the space dimensions: these rays of light are really emanating out in a sphere around me. Because light travels at 45 degrees, anything traveling slower than light from this t=0 event is closer to the time axis than the light rays, and anything faster than light is further away from the time axis.
The light rays define the future lightcone. This is the set of spacetime events that can perceive the event at t=0, and so, in a Universe without FTL, all the events that can be affected by whatever happened at this event at t=0. There is also a past lightcone, which would be the 45 degree lines extending backwards in time from the event: in a Universe without FTL this defines all the events that could have effected that t=0 event, because the light (and thus things moving slower than light) from those other events had time to reach the t=0 event.
So now let me move from general spacetime diagrams to an example that will indicate why FTL implies time travel. Let's consider a specific example: Let's say we on Earth have built a FTL communication device that let's us talk to the inhabitants of the planet Proxima Centauri B, 4.25 lightyears away. Again, this is what a FTL and slower-than-light set of communications would appear like in a spacetime diagram.
Critically, I've drawn the time axis for the Proximal Centaurians parallel to our own time. This is because Proxima Centauri is moving at essentially the same velocity as Earth (the differences are small compared to the speed of light). Thus, there are no big relativistic effects between our counting of time and the Proximal Centaurians.
Now, let's imagine that some event occurs away from Earth, oriented in such a way that the light from the event hits us before it reaches Proxima Centauri. The spacetime diagram for that would look like what I've shown. First we see the light, then the light reaches Proxima Centauri. Notice I've drawn the light rays from the event traveling at 45 degrees to my time axis. After all, it is light, and light travels at 45 degrees on spacetime diagrams.
So, now, let's add in FTL communication. We see the event, we get on the FTL phone, and we tell the Proximal Centaurians. They get the phone call, and now have years to prepare for the arrival of the light from whatever the event is (let's say it's a supernovae, or the launch of relativistic attack vehicles. We are playing with sci-fi tropes here).
Now, this is the image most people have of FTL communication. There appears to be no problem: we all agree that the event happened "first," then Earth calls Proxima Centauri, then the light reaches Proxima Centauri. No problem: though the Proximal Centaurians hear about the event "early," no causality has been violated. After all, we all agree on what happened first, don't we? No effect precedes its cause.
But we've forgotten relativity. This only works when everyone is moving in the same frame of reference, like us and Proxima Centauri (really we're not in the same rest frame, but its close enough not to matter). So, to see the problem, let's add a new observer, moving at high speeds relative to Earth and Proxima Centauri. It's sci-fi, so we add a relativistic spaceship. It's moving with v<c but v>>0, so it's trajectory on my spacetime diagram is highly skewed relative to my time axis: it's nearly moving at the speed of light.
Here's where the relativistic effects start coming into play. Relativity tells us that everyone moving with constant velocity is totally justified in saying they are stationary. Thus, we think we're stationary (ignore the rotation of the Earth, or its orbit around the Sun). The Proximal Centaurians think they're at rest. The people on the relativistic spaceship think they are at rest. So they can draw their spacetime diagram, with their own time axis. That time axis, like mine, is always where they are: on the ship. So I think their time axis is aligned with the ship trajectory.
In addition, they have a space-axis, just like I do. Relativity mixes up space and time, so their space axis I perceive as slanted - just like their time axis is skewed. It turns out that the space axis is flipped across the 45 degree null line, but I'm not going to prove that. This weird mixture of space and time of observers I perceive as moving is a necessary part of relativity. It is the only way everyone can agree that light moves at c.
Now, if I wanted to, I could draw the spacetime diagram of the spaceship in its frame of reference. It would have othogonal space and time axes, the light from the event would travel at 45 degree, and they would see Earth's axes highly skewed (pointing toward the left if I kept the same orientation as in this set of diagrams). That's relativity. But we will not draw that diagram here, as it's not necessary for the story.
So what happens now. Let's ask when the spaceship sees the various events in this diagram. To do that, we need to know the lines of constant time for the ship. That's not too hard: lines of constant time for us are lines in the spacetime diagram parallel to the space axis. So it is for the spaceship. Their lines of constant time look like this:
Now do you see the problem?
According to us, on Earth, the order of events is thus: we see the light from the event hit us. We call Proxima Centauri on the FTL phone. The Proximal Centaurians do whatever they want to do in response to that call, and then they see the light of the event.
What does the ship see? They see the phone call received on Proxima Centauri. Then they see the phone call placed from Earth. Effect precedes cause: causality is violated. In fact, if the ship had a FTL phone set up in the right way, they could call Earth before Earth placed the call. They could even tell Earth "hey, don't make that call to Proxima Centauri we just saw you make." Then what?
Now, you might say "wait, light takes a finite amount of time to travel. You've just shown what times the spaceship will assign to various events, but they can't see it immediately. That'll save us!" Sadly no. Here's when the ship actually gets the light from the events.
As you can see, the light from the phone call reception arrives well before the light from the placing of the phone call. Again: causality is violated.
So you see the problem: if we could just say that there was only one frame of reference where we needed to set up cause and effect, then we could have FTL without worrying about causality. However, there is no special frame of reference, there cannot be one if relativity is to be true. And relativity is true, because we all measure light to travel at the same speed (also, you need relativity for electromagnetism to work, which you probably do want).
If you could travel or communicate FTL, you can time travel, or at least communicate backwards in time. And that would be troubling - and doesn't seem to be the Universe we live in.